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Optimization
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Anisotropic best τ C -approximation in
normed spaces
a
b
Xian-Fa Luo , Chong Li & Jen-Chih Yao
c
a
Department of Mathematics, China Jiliang University, Hangzhou
310018, P.R. China
b
Department of Mathematics, Zhejiang University, Hangzhou
310027, P.R. China
c
Department of Applied Mathematics, National Sun Yat-sen
University, Kaohsiung 804, Taiwan
Available online: 03 Nov 2010
To cite this article: Xian-Fa Luo, Chong Li & Jen-Chih Yao (2011): Anisotropic best τ C approximation in normed spaces, Optimization, 60:6, 725-738
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Optimization
Vol. 60, No. 6, June 2011, 725–738
Anisotropic best sC -approximation in normed spaces
Xian-Fa Luoa*, Chong Lib and Jen-Chih Yaoc
a
Department of Mathematics, China Jiliang University, Hangzhou 310018,
P.R. China; bDepartment of Mathematics, Zhejiang University, Hangzhou 310027,
P.R. China; cDepartment of Applied Mathematics, National Sun Yat-sen University,
Kaohsiung 804, Taiwan
Downloaded by [Zhejiang University] at 00:04 24 December 2011
(Received 7 June 2009; final version received 30 July 2010)
The notions of the C -Kolmogorov condition, the C -sun and the
C -regular point are introduced, and the relationships between them and
the best C -approximation are explored. As a consequence, characterizations of best C -approximations from some kind of subsets (not necessarily
convex) are obtained. As an application, a characterization result for a set
C to be smooth is given in terms of the C -approximation.
Keywords: Minkowski function; the best C -approximation;
C -Kolmogorov condition; the C -sun; smoothness
the
AMS Subject Classification: 41A65
1. Introduction
Let X be a real normed linear space and C be a closed, bounded, convex subset of X
having the origin as an interior point. Recall that the Minkowski function pC with
respect to the set C is defined by
pC ðxÞ ¼ infft 4 0 : x 2 tCg 8 x 2 X:
Let G be a subset of X and x 2 X. Following [7], define the minimal time function
C ð; GÞ by
C ðx; GÞ :¼ inf pC ð g xÞ 8 x 2 X:
g2G
ð1:1Þ
The study of the minimal time function C ð; G Þ is motivated by its worldwide
applications in many areas of variational analysis, optimization, control theory,
approximation theory, etc., and has received a lot of attention; see e.g.
[5–7,9,11–13,17,18]. In particular, the proximal subgradient of the minimal time
function C ð; GÞ is estimated and computed in [6,17,18], for Hilbert spaces with
applications to control theory; while other various subgradients, such as the Fréchet
subgradients, Clarke subgradients, the -subdifferential as well as the limiting
*Corresponding author. Email: luoxianfaseu@163.com
ISSN 0233–1934 print/ISSN 1029–4945 online
ß 2011 Taylor & Francis
DOI: 10.1080/02331934.2010.523110
http://www.informaworld.com
726
X.-F. Luo et al.
subdifferential of the minimal time function C ð; GÞ in Hilbert spaces and/or general
Banach spaces are explored in [5,9,13].
Our interest in this article is focused on the following minimization problem,
denoted by min(x, G ),
min pC ð g xÞ,
ð1:2Þ
g2G
where x 2 X. Clearly, g0 2 G is a solution of the problem min(x, G) if and only if
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pC ð g0 xÞ ¼ C ðx; GÞ:
According to [7], any solution of the problem min(x, G) is called a best
C -approximation (or, generalized best approximation) to x from G. We denote
by PC
G ðxÞ the set of all best C -approximations to x from G. The generic wellposedness of the minimization problem min(x, G ) in terms of the Baire category was
studied in [7,11], while the relationships between the existence of solutions and
directional derivatives of the function C(x, G ) was explored in [12].
In the special case when C is the closed unit ball B of X, the minimal time
function (1.1) and the corresponding minimization problem (1.2) are reduced to the
distance function of C and to the classical best approximation, respectively, which
has been studied extensively and deeply, see, e.g. [2,16,19].
One aim of this article is to characterize the class of subsets of X for which the
so-called C -Kolmogorov condition holds about best C -approximations. Another
aim of this article is to prove the equivalence between the smoothness of the
underlying set C and the convexity of C -B-suns. In particular, by taking C to be the
closed unit ball of X, our results extend the corresponding ones for nonlinear
approximation problems; see, e.g. [2,4,19].
2. Preliminaries
Let X be a real normed linear space and let X* denote its topological dual. Let A be a
nonempty subset of X. As usual, we use bd A and int A to denote the boundary and
the interior of A, respectively. The polar of A is denoted by A and defined by
A ¼ fx 2 X : x ðxÞ 1 8 x 2 Ag:
Then A is a weakly* closed convex subset of X*. Furthermore, in the case when A is
a convex bounded set with 0 2 int A, A is weakly* compact with 0 2 int A .
In particular, B equals the closed unit ball of X*. Moreover, for a set A X , ext A
and A stand for the set of all extreme points and the weak*-closure of A,
respectively. The following proposition is exactly the well-known Krein–Milman
theorem, see, e.g. [10].
PROPOSITION 2.1 Suppose that A is a compact convex subset of X*. Then A equals the
closed convex closure of ext A.
Let
¼ inf pC ðxÞ
kxk¼1
and ¼ sup pC ðxÞ:
kxk¼1
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Optimization
We end this section with some known and useful properties of the Minkowski
function; see [15, Section 1] for assertions (i)–(v) while (vi) is an immediate
consequence of (iii) and (v).
PROPOSITION 2.2
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(i)
(ii)
(iii)
(iv)
(v)
(vi)
Let x, y 2 X and x 2 X . Then we have the following assertions.
pC ðxÞ 0, and pC ðxÞ ¼ 0()x ¼ 0.
pC ðx þ yÞ pC ðxÞ þ pC ð yÞ.
pC ðxÞ ¼ pC ðxÞ for each 4 0.
pC ðxÞ 1()x 2 C.
pC ðxÞ ¼ supx 2C x ðxÞ and pC ðx Þ ¼ supx2C x ðxÞ.
kxk pC ðxÞ kxk.
3. Characterization of the best qC -approximation
The notion of suns introduced by Efimov and Stechkin (cf. [8]) has proved to be
rather important in nonlinear approximation theory; see, e.g. [2–4,8,19] and
references therein. In the following definition we extend this notion to the case of
the generalized approximation. Throughout this article, we always assume that
G X is a nonempty subset of X, and let x 2 X and g0 2 G, unless specially stated.
Definition 3.1
The element g0 is called
C
(a) a C -solar point of G with respect to x if g0 2 PC
G ðxÞ implies that g0 2 PG ðx Þ
for each 4 0, where x ¼ g0 þ ðx g0 Þ;
(b) a C -solar point of G if g0 is a C -solar point of G with respect to each x 2 X.
We say that G is a C -sun of X if each point of G is a C -solar point of G.
Remark 3.1
We always write
x ¼ g0 þ ðx g0 Þ
8 40
ð3:1Þ
if no confusion caused. Thus
pC ð g0 x Þ ¼ pC ð g0 xÞ
8 4 0:
ð3:2Þ
Furthermore, the following implication holds (cf. [13, Lemma 3.1]):
C
g0 2 PC
G ðxÞ ¼) g0 2 PG ðx Þ 8 2 ½0, 1:
ð3:3Þ
For two points x, y 2 X, we use [x, y] to denote the closed interval with ends x
and y, that is,
½x, y :¼ ftx þ ð1 tÞ y : t 2 ½0, 1g:
Recall (cf. [1]) that g0 2 G is a star-shaped point of G if ½ g0 , g G for each g 2 G. If G
has a star-shaped point g0, then G is called a star-shaped set with vertex g0. Clearly,
a convex subset is a star-shaped set with each vertex g0 2 G. We use S(g0, G) to denote
the star-shaped set with vertex g0 generated by G, that is,
[
Sð g0 , G Þ :¼
½ g0 , g:
g2G
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X.-F. Luo et al.
PROPOSITION 3.1 Suppose that g0 2 G is a star-shaped point of G. Then g0 is a
C -solar point of G. Consequently, any convex subset of X is a C -sun of X.
Proof Let x 2 X be such that g0 2 PC
G ðxÞ. By Remark 3.1, we only need to show that
g0 2 P C
ðx
Þ
for
each
4
1.
To
this
end,
let 4 1 and g 2 G. Since g0 is a star-shaped
G point of G, it follows from Proposition 2.2 (iii) that
1
1
1
1
g0 þ g x ¼ pC ð g x Þ:
pC ð g0 xÞ pC
By (3.2),
pC ð g0 x Þ ¼ pC ð g0 xÞ pC ð g x Þ:
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Hence, g0 2 PC
G ðx Þ. This shows that g0 is a C -solar point of G.
g
The following proposition gives an equivalent condition for C -solar points in
terms of star-shaped points.
PROPOSITION 3.2
The element g0 is a C -solar point of G if and only if
C
g0 2 PC
G ðxÞ () g0 2 PSð g0 , GÞ ðxÞ 8 x 2 X:
C
Proof For x 2 X, it is clear that g0 2 PC
Sð g0 , G Þ ðxÞ ¼) g0 2 PG ðxÞ. Thus, to complete the
proof, it suffices to verify that g0 is a C -solar point of G if and only if
C
g0 2 PC
G ðxÞ ¼) g0 2 PSð g0 , GÞ ðxÞ
ð3:4Þ
holds for each x 2 X. To this end, let x 2 X. By Definition 3.1 and Remark 3.1, one
has that g0 is a C -solar point of G if and only if the following implication holds:
C
1
8 2 ½0, 1Þ:
ð3:5Þ
g0 2 PC
G ðxÞ ¼) g0 2 PG x1
Note by Proposition 2.2 (iii) that
1
, 8 2 ½0, 1Þ
g0 2 PC
G x1
() pC ð g0 xÞ pC ððg0 þ ð1 Þ gÞ xÞ 8 g 2 G, 8 2 ½0, 1Þ
() g0 2 PC
Sð g0 , GÞ ðxÞ:
Hence, (3.5) holds if and only if (3.4) holds. This completes the proof.
g
Definition 3.2 The element g0 is called a local best C -approximation to x from G if
there exists an open neighbourhood U(g0) of g0 such that g0 2 PC
G\Uð g0 Þ ðxÞ.
Clearly, if g0 2 PC
G ðxÞ, then g0 is a local best C -approximation to x from G.
The following proposition shows that the converse remains true if g0 is a C -solar
point of G.
PROPOSITION 3.3 Suppose that g0 is a C -solar point of G. Then g0 2 PC
G ðxÞ if and only
if g0 is a local best C -approximation to x from G.
Proof The necessity part is clear as noted earlier. Below we prove the sufficiency
part. To this end, suppose that g0 is a local best C -approximation to x from G. Then
there exists an open neighbourhood U(g0) of g0 such that g0 2 PC
Uð g0 Þ\G ðxÞ. We claim
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Optimization
that there is 4 0 such that g0 2 PC
G ðx Þ. Indeed, otherwise, one has that
ðx
Þ
for
each
n
2
N.
Thus
there
exists a sequence fgn g G such that,
g0 6 2 PC
1=n
G
for each n 2 N,
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1
pC ð gn x1=n Þ 5 pC ð g0 x1=n Þ ¼ pC ð g0 xÞ:
n
ð3:6Þ
It follows from Proposition 2.2 (vi) that limn!1 gn ¼ g0 . This implies that there
exists n0 2 N such that gn0 2 Uð g0 Þ \ G. This together with (3.6) implies that
C
g0 6 2 PC
Uð g0 Þ\G ðx1=n0 Þ, which is a contradiction by Remark 3.1 as g0 2 PUð g0 Þ\G ðxÞ.
C
Hence the claim stands; that is g0 2 PG ðx Þ for some 4 0. Noting that
x ¼ g0 þ 1 ðx g0 Þ and that g0 is a C -solar point of G, we have that
1
C
g0 2 PG g0 þ ðx g0 Þ ¼ PC
G ðxÞ
g
and completes the proof.
For the sequel study, we need to introduce the following notation:
g0 x :¼ fx 2 C : x ð g0 xÞ ¼ pC ð g0 xÞg
Then g0 x is a nonempty, weakly* compact convex subset of C . Furthermore,
write E g0 x :¼ extg0 x . Then, E g0 x 6¼ ; by Proposition 2.1. Moreover, by
definition, we have that
E g0 x ¼ extC \ g0 x :
ð3:7Þ
The notions stated in Definition 3.3 are extension of the notions of Kolmogorov
condition and Papini condition in approximation theory to the setting of the best
C -approximation theory, see, e.g. [3,4,19] and [14].
Definition 3.3
The pair (x, g0) is said to satisfy
(a) the C -Kolmogorov condition (the C -KC, for short) if
max x ð g g0 Þ 0
x 2g0 x
8 g 2 G,
ð3:8Þ
(b) the C -Papini condition (the C -PC, for short) if
max x ð g0 gÞ 0 8 g 2 G:
x 2gx
ð3:9Þ
Clearly, by (3.7), g0 x in (3.8) and gx in (3.9) can be replaced by E g0 x and
E gx , respectively.
In the case when C is the closed unit ball of X, that is, pC is the norm, the notions
of the regular point and the strongly regular point were introduced and studied in [4]
and [19], respectively. The following notions of the C -regular point and the strongly
C -regular point are, respectively, generalizations of the corresponding regular point
and strongly regular point in best approximation theory.
Definition 3.4
The element g0 is called
(a) a C -regular point of G with respect to x if for any weakly* closed subset A
of C satisfying for some g 2 G the condition
g0 x A
and
min
x ð g0 gÞ 4 0,
x 2A
ð3:10Þ
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X.-F. Luo et al.
there exists a sequence fgn g G such that gn ! g0 and
x ð g0 gn Þ 4 x ð g0 xÞ pC ð g0 xÞ 8 x 2 A, 8 n 2 N;
ð3:11Þ
(b) a strongly C -regular point of G with respect to x if for any weakly* closed
subset A of C satisfying (3.10) for some g 2 G, there exists a sequence
fgn g G such that gn ! g0 and
x ð g0 gn Þ 4 0 8 x 2 A,
8 n 2 N:
ð3:12Þ
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We say that G is a C -regular set (resp. strongly C -regular set) of X if each point
of G is a C -regular point (resp. strongly C -regular point) of G with respect to
each x 2 X.
Roughly speaking, a C -regular point g0 of G with respect to x means that, for
any weakly* closed neighbourhood A of the set g0 x , if there exists some g 2 G such
that g0 g is positive on A, then any neighbourhood of g0 contains an element g0 2 G
such that g0 g0 has the same sign as g0 g on the set g0 x while out of this set it
can have opposite sign but have to be controlled within pC ð g0 xÞ x ð g0 xÞ;
while, for a strongly C -regular point g0, g0 g0 , must have the same sign as g0 g on
the whole weakly* closed neighbourhood A.
Remark 3.2 Clearly, the strong C -regularity implies the C -regularity. We do
not know whether the converse is true even in the case when C is the closed unit
ball of X.
Remark 3.3 Clearly, any interior point of G is a strongly C -regular point of G.
Moreover, it is not difficult to check that any star-shaped point of G is a strongly
C -regular point of G.
Below we provide an example to illustrate these notions.
Example 3.1 Let X :¼ R2 be the two-dimensional
Euclidean
pffiffiffi
pffiffiffi space. Let C be the
equilateral triangle with the vertexes ð0, 2Þ, ð 3, 1Þ and ð 3, 1Þ (Figure 1). Then
with vertexes x1 , x2 and x3 , where x1 :¼
0p2ffiffi int C and C is the equilateral triangle
pffiffi
3 1
3 1
ð 2 , 2Þ, x2 :¼ ð0, 1Þ and x3 :¼ ð 2 , 2Þ, and so ext C ¼ fx1 , x2 , x3 g (Figure 2).
t2
(0,2)
t1
√
(− 3,−1)
Figure 1. The closed convex set C.
√
( 3,−1)
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Optimization
t2
√
√
(− 23, 12)
( 23, 21)
t1
(0,−1)
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Figure 2. The polar C of C.
t2
G
t1
t2 = 12t1
t2 =− 12t1
Figure 3. The set G.
Let G be the subset defined by
[
1
1
ðt1 , t2 Þ : t2 t1 , t1 0
G ¼ ðt1 , t2 Þ : t2 t1 , t1 0
2
2
(Figure 3). Clearly, G is not convex. We shall show that G is a strongly C -regular
set. To this end, let g0 2 G. By Remark 3.3, we assume, without loss of generality,
that g0 ¼ ða1 , a2 Þ 2 bd G satisfies that
1
ð3:13Þ
a1 4 0 and a2 ¼ a1
2
(noting that the origin is a star-shaped point of G ). Let A be a weakly* closed subset
of C satisfying (3.10) for some g ¼ ðb1 , b2 Þ 2 G and some x 2 X. In the case when
b1 0 and b2 12 b1 , choose gn ¼ ð1 1nÞ g0 þ 1n g 2 G for each n 2 N. Then it is easy
to see that fgn g is as desired. Below we consider the case when b1 0 and b2 12 b1 .
It follows from (3.13) that
1
1
ða2 b2 Þ ða1 b1 Þ ¼ a1 þ b1 b2 a1 5 0:
2
2
Consequently,
pffiffiffi
pffiffiffi
3
3
1
1
ða1 b1 Þ þ ða2 b2 Þ ða1 b1 Þ þ ða1 b1 Þ
x3 ð g0 gÞ ¼ 2 pffiffiffi
2
2
4
12 3
ða1 b1 Þ 5 0:
¼
4
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X.-F. Luo et al.
This means that x3 6 2 A. Therefore, it suffices to consider the case when A ¼ fx1 , x2 g.
To this end, letpffiffign ¼ ð1 1nÞ g0 for each n 2 N. Then fgn g G and gn ! g0 . Noting
that x1 ð g0 Þ ¼ ð 23 14Þa1 and x2 ð g0 Þ ¼ 12 a1 , we have that
1
1
min x ð g0 gn Þ ¼ min
x ð g0 Þ ¼ a1 4 0
n x 2A
2n
x 2A
8 n 2 N:
This means that g0 is a strongly regular point of G, and so G is a strongly
C -regular set.
THEOREM 3.1
Consider the following assertions:
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(i) The pair (x, g0) satisfies the C -KC.
(ii) The point g0 2 PC
G ðxÞ.
(iii) The pair (x, g0) satisfies the C -PC.
Then (i) ) (ii) ) (iii). In addition, if g0 is a strongly C -regular point of G with
respect to x, then (i) , (ii) , (iii).
Proof (i) ) (ii). Suppose that (i) holds and let g 2 G. Then by Definition 3.3 (a),
there exists x 2 g0 x such that x ð g g0 Þ 0. This together with Proposition 2.2
(v) implies that
pC ð g0 xÞ ¼ x ð g0 gÞ þ x ð g xÞ x ð g xÞ pC ð g xÞ;
hence, g0 2 PC
G ðxÞ as g 2 G is arbitrary and (ii) is proved.
(ii) ) (iii). Suppose that (ii) holds. Let g 2 G and x 2 gx . Then x ð g xÞ ¼
pC ð g xÞ; hence,
x ð g0 gÞ ¼ x ð g0 xÞ þ x ðx gÞ pC ð g0 xÞ pC ð g xÞ 0:
This shows that (x, g0) satisfies the C -PC and (iii) is proved.
Suppose that g0 is a strongly C -regular point of G with respect to x. It suffices to
prove the implication (iii) ) (i). To this end, suppose on the contrary that (x, g0) does
not satisfy the C -KC. Then there exist g 2 G and 4 0 such that
max x ð g g0 Þ ¼ :
ð3:14Þ
x 2g0 x
Let
U ¼ x 2 C : x ð g g0 Þ 5 2
and
A¼U :
ð3:15Þ
Then A is a weakly* closed subset of C satisfying g0 x A and
min
x ð g0 gÞ x 2A
4 0:
2
ð3:16Þ
Since g0 is a strongly C -regular point of G with respect to x, there exists a sequence
fgn g G such that
lim gn ¼ g0
n!1
ð3:17Þ
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Optimization
and
min
x ð g0 gn Þ 4 0
x 2A
8 n 2 N:
ð3:18Þ
Let K ¼ C n U. Then K is the weakly* compact subset of C . Moreover, we have
that K \ g0 x ¼ ; because g0 x U by (3.14) and (3.15). Consequently,
:¼ max
x ð g0 xÞ 5 pC ð g0 xÞ:
ð3:19Þ
1
0 :¼ ð pC ð g0 xÞ Þ:
2
ð3:20Þ
x 2K
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Set
As limn!1 gn ¼ g0 by (3.17), one has from Proposition 2.2 (vi) that there exists
n0 2 N such that
maxfpC ð gn0 g0 Þ, pC ð g0 gn0 Þg 5 0 :
ð3:21Þ
pC ð g0 xÞ pC ð g0 gn0 Þ þ pC ð gn0 xÞ 5 0 þ pC ð gn0 xÞ:
ð3:22Þ
Thus
It follows from Proposition 2.2 (v), (3.19)–(3.22) that
max
x ð gn0 xÞ max
x ð gn0 g0 Þ þ max
x ð g0 xÞ
x 2K
x 2K
x 2K
pC ð gn0 g0 Þ þ 5 0 þ ¼ pC ð g0 xÞ 0
5 pC ð gn0 xÞ:
This shows that K \ gn0 x ¼ ;, and so gn0 x C n K ¼ U. Therefore,
max x ð g0 gn0 Þ inf
x ð g0 gn0 Þ ¼ min
x ð g0 gn0 Þ 4 0
x 2gn
0 x
x 2U
x 2A
thanks to (3.18). This contradicts the C -PC for the pair (x, g0), and the proof is
complete.
g
Remark 3.4 One natural and interesting question is: whether the implication
(iii) ) (i) remains true under simple (not strong) regularity assumption? Even in the
case when C is the unit closed ball, we do not know the answer and so we leave
it open.
The main result of this section is a generalization of [3, Theorem 9].
THEOREM 3.2
The following assertions are equivalent.
(i) The element g0 is a C -solar point of G with respect to x.
(ii) The element g0 2 PC
G ðxÞ if and only if (x, g0) satisfies the C -KC.
(iii) The element g0 is a C -regular point of G with respect to x.
Proof (i) ) (ii). Suppose that (i) holds. The sufficiency part of (ii) follows directly
from Theorem 3.1. Below we show the necessity part of (ii). To this end, we assume
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X.-F. Luo et al.
that g0 2 PC
G ðxÞ. Without loss of generality, we may assume that x 6¼ g0 and
pC ð g0 xÞ ¼ 1. Let g 2 G n fg0 g be arbitrary. Then g0 2 PC
½ g0 , g ðxÞ by Proposition 3.2
and ½ g0 , g \ intðC þ xÞ ¼ ;. Thus, by the separation theorem (cf. [10]), there exist
y 2 X n f0g and a real number r such that
y ðz xÞ r 8 z 2 ½ g0 , g
ð3:23Þ
y ð y xÞ r 8 y 2 C þ x:
ð3:24Þ
and
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In particular, we have that pC ð y Þ r by Proposition 2.2 (v). Let x ¼ r1 y . Then
pC ðx Þ 1 and so x 2 C . Since g0 2 ½ g0 , g \ ðC þ xÞ, it follows from (3.23)
and (3.24) that
r ¼ y ð g0 xÞ:
ð3:25Þ
This implies that x ð g0 xÞ ¼ 1 ¼ pC ð g0 xÞ, and so x 2 g0 x . Furthermore,
x ð g g0 Þ 0 thanks to (3.23) and (3.25). Hence the necessity part holds as
g 2 G n fg0 g is arbitrary and the implication is proved.
(ii) ) (i). Suppose that (ii) holds and assume that g0 2 PC
G ðxÞ. Then
max x ð g g0 Þ 0
x 2g0 x
8 g 2 G:
ð3:26Þ
Let 4 0 be arbitrary. Noting that g0 x ¼ ð g0 xÞ, one has that
g0 x ¼ g0 x . This together with (3.26) implies that ðx , g0 Þ satisfies the C -KC,
and so g0 2 PC
G ðx Þ. This shows that g0 is a C -solar point of G with respect to x.
(ii) ) (iii). Suppose that (ii) holds. Let g 2 G and A be a weakly* closed subset of
C satisfying (3.10). Then
x ð g g0 Þ 5 0:
max x ð g g0 Þ max
x 2g0 x
x 2A
This implies that (x, g0) does not satisfy the C -KC; hence, g0 6 2 PC
G ðxÞ by (ii). Since g0
is a C -solar point of G with respect to x by the equivalence of (i) and (ii) just proved,
it follows from Proposition 3.3 that g0 is not local best C -approximation to x from
G. Thus there exists a sequence fgn g G such that gn ! g0 and
pC ð gn xÞ 5 pC ð g0 xÞ
8 n 2 N:
ð3:27Þ
Let x 2 A be arbitrary. Then, x ð gn xÞ pC ð gn xÞ for each n 2 N. This and
(3.27) imply that
x ð g0 gn Þ x ð g0 xÞ pC ð gn xÞ 4 x ð g0 xÞ pC ð g0 xÞ 8 n 2 N:
In view of Definition 3.4, g0 is a C -regular point of G.
(iii) ) (ii). Suppose that (iii) holds. By Theorem 3.1 (i), it suffices to prove that
(x, g0) satisfies the C -KC whenever g0 2 PC
G ðxÞ. To this end, we assume on the
contrary that it is not the case. Then, there are g 2 G and 4 0 such that (3.14) holds.
Let U and A be defined by (3.15). Then (3.16) holds. Since g0 is a C -regular point of
G with respect to x, there exists a sequence fgn g G such that limn!1 gn ¼ g0
and (3.11) holds. This together with Proposition 2.2 implies that
x ð gn xÞ 5 pC ð g0 xÞ 8 n 2 N:
max
x 2A
ð3:28Þ
735
Optimization
On the other hand, let K ¼ C n U. Then there is 4 0 such that
max
x ð g0 xÞ 5 pC ð g0 xÞ :
x 2K
ð3:29Þ
Since limn!1 gn ¼ g0 , one has that limn!1 pC ð gn g0 Þ ¼ 0. Let n0 2 N be such that
pC ð gn0 g0 Þ 5 . It follows from (3.29) that
max
x ð gn0 xÞ pC ð gn0 g0 Þ þ max
pC ð g0 xÞ 5 pC ð g0 xÞ:
x 2K
x 2K
ð3:30Þ
Combining (3.28) and (3.30), we obtain that pC ð gn0 xÞ 5 pC ð g0 xÞ, which
contradicts that g0 2 PC
g
G ðxÞ. The proof is complete.
The following corollary is a global version of Theorem 3.2.
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COROLLARY 3.1
The following statements are equivalent.
(i) G is a C -sun of X.
(ii) For each g0 2 G and each x 2 X, g0 2 PC
G ðxÞ if and only if (x, g0) satisfies the
C -KC.
(iii) G is a C -regular set.
4. Smoothness and convexity of qC -B-suns
We begin with the notion of smooth convex sets (cf. [10]). Let x 2 bd C and x 2 C .
Recall that x* is a supporting functional of C at x if x ðxÞ ¼ 1.
Definition 4.1 The set C is called smooth if each point of bd C has a unique
supporting functional.
The following notion extends a similar concept introduced in [2].
Definition 4.2 The set G is called a C -B-sun of X if for each x 2 X there exists
g0 2 P C
G ðxÞ such that g0 is a C -solar point of G with respect to x.
Remark 4.1 Clearly, if G is a existence set (i.e. PC
G ðxÞ 6¼ ; for each x 2 X), then a
C -sun must be C -B-sun. The converse is not true in general, see [19, Example 1.4]
for the case when C is the unit ball of X.
The main result of this section is as follows, which extends [2, Theorem 2.5] to the
setting of the best C -approximation.
THEOREM 4.1
The set C is smooth if and only if each C -B-sun of X is convex.
Proof ‘ ) ’. Suppose that C is smooth and G is a C -B-sun of X. It suffices to verify
that 12 ð g1 þ g2 Þ 2 G for each pair of elements g1 , g2 2 G. To this end, let g1 , g2 2 G and
let x ¼ 12 ð g1 þ g2 Þ. By Definition 4.2 there exists g0 2 PC
G ðxÞ such that g0 is a C -solar
point of G with respect to x. It follows from Theorem 3.2 that there exist
x1 , x2 2 g0 x such that
xi ð gi g0 Þ 0 8 i ¼ 1, 2:
ð4:1Þ
Suppose that g0 6¼ x and consider the point y :¼ ð g0 xÞ=pC ð g0 xÞ. Then y 2 C
¼ 1 for each i ¼ 1, 2. Noting that each xi 2 C , we have that xi , i ¼ 1, 2,
and xi ðyÞ
Hence, x1 ¼ x2 by the smoothness of C.
are supporting functionals of C at y.
Let x ¼ x1 .
736
X.-F. Luo et al.
Then
1
1
pC ð g0 xÞ ¼ x ð g0 xÞ ¼ x ð g0 g1 Þ þ x ð g0 g2 Þ 0
2
2
thanks to (4.1). By Proposition 2.2 (i), one has x ¼ g0 , which is a contradiction, and
hence 12 ð g1 þ g2 Þ 2 G.
‘(’. Conversely, suppose on the contrary that C is not smooth. Then there exist
x0 2 bd C and two functionals x1 , x2 2 C such that
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x1 6¼ x2
and
x1 ðx0 Þ ¼ x2 ðx0 Þ ¼ 1:
ð4:2Þ
Let Gi :¼ fx 2 X : xi ðxÞ 0g for each i ¼ 1, 2, and let G ¼ G1 [ G2 . Then G is a closed
and nonconvex subset of X. In fact, the closedness is clear. To prove the
nonconvexity, we first prove that kerðx1 Þ n G2 6¼ ;, where kerðx Þ :¼
fx 2 X : x ðxÞ ¼ 0g is the kernel of the functional x*. Indeed, otherwise, one has
that kerðx1 Þ G2 . Let x 2 X. Then, by (4.2),
x xi ðxÞx0 2 kerðxi Þ 8 i ¼ 1, 2:
ð4:3Þ
In particular, we have that x x1 ðxÞx0 2 G2 . It follows that
x2 ðxÞ x2 ðx0 Þx1 ðxÞ ¼ x1 ðxÞ 8 x 2 X:
6 ;.
This implies that x1 ¼ x2 , which is a contradiction. Therefore, kerðx1 Þ n G2 ¼
Similarly, we also have that kerðx2 Þ n G1 6¼ ;. Take x1 2 kerðx1 Þ n G2 and
x2 2 kerðx2 Þ n G1 . Then x1 , x2 2 G and xi ð12 x1 þ 12 x2 Þ 5 0 for each i ¼ 1, 2. This
means that 12 x1 þ 12 x2 6 2 G, and so G is not convex.
By the definition of C, it is easy to see that
C ðx; GÞ ¼ minfC ðx; G1 Þ, C ðx; G2 Þg
8 x 2 X:
ð4:4Þ
We will prove that, for each i ¼ 1, 2,
C ðx; Gi Þ ¼ xi ðxÞ
8 x 2 X n Gi :
ð4:5Þ
To this end, fix i ¼ 1, 2 and let x 2 X n Gi . Then xi ðxÞ 5 0, and by (4.3), one has that
C ðx; Gi Þ pC ððx xi ðxÞx0 Þ xÞ ¼ xi ðxÞ pC ðx0 Þ ¼ xi ðxÞ
(noting that pC ðx0 Þ ¼ 1). On the other hand,
C ðx; Gi Þ ¼ inf pC ð g xÞ inf xi ð g xÞ xi ðxÞ,
g2Gi
g2Gi
and the assertion (4.5) is seen to hold.
Below we prove that G is a C -B-sun of X. To this end, let x 2 X n G and
d :¼ C ðx; GÞ. Without loss of generality, we may assume that
C ðx; G1 Þ C ðx; G2 Þ:
ð4:6Þ
d ¼ C ðx; G1 Þ ¼ x1 ðxÞ:
ð4:7Þ
Then by (4.5),
Optimization
737
g0 ¼ x x1 ðxÞx0 2 kerðx1 Þ G1 G:
ð4:8Þ
pC ð g0 xÞ ¼ dpC ðx0 Þ ¼ d ¼ C ðx; G1 Þ ¼ C ðx; GÞ:
ð4:9Þ
Let g0 ¼ x þ dx0 . Then by (4.3)
Moreover
g0 2 PC
G ðxÞ.
g0 2 PC
G ðx Þ
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We assert that
for each 4 0. Granting this, G is a
Hence
nonconvex, C -B-sun and the proof of Theorem 4.1 is complete. Let 4 0. By (4.8)
and (4.9), we have that g0 2 PC
G1ðxÞ. Since G1 is convex, it follows from Proposition 3.1
C
that g0 2 PG1 ðx Þ. Thus, to complete the proof, it suffices to show that
C ðx ; GÞ ¼ C ðx ; G1 Þ. To do this, note that x2 ðxÞ x1 ðxÞ ¼ d by (4.5), (4.6) and
(4.7). Consequently,
x2 ðx Þ ¼ ð1 Þx2 ð g0 Þ þ x2 ðxÞ
¼ ð1 Þðx2 ðxÞ þ dx2 ðx0 ÞÞ þ x2 ðxÞ
¼ x2 ðxÞ þ d ð1 Þ
d þ d ð1 Þ
¼ d:
This together with (4.5) (with x in place of x) implies that
C ðx ; G2 Þ d ¼ C ðx ; G1 Þ:
Therefore, C ðx ; GÞ ¼ C ðx ; G1 Þ thanks to (4.4). The proof is complete.
g
Acknowledgements
X.-F. Luo was supported in part by the NNSF of China (Grant No. 90818020) and the NSF
of Zhejiang Province of China (Grant No. Y7080235), C. Li is supported in part by the NNSF
of China (Grant Nos. 10671175, 10731060) and J.-C. Yao is supported by the grant NSC
97-2115-M-110-001 (Taiwan).
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